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Homotopy principles for equivariant isomorphisms


Authors: Frank Kutzschebauch, Finnur Lárusson and Gerald W. Schwarz
Journal: Trans. Amer. Math. Soc. 369 (2017), 7251-7300
MSC (2010): Primary 32M05; Secondary 14L24, 14L30, 32E10, 32M17, 32Q28
DOI: https://doi.org/10.1090/tran/6797
Published electronically: May 5, 2017
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Abstract: Let $ G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $ X$ and $ Y$. Let $ p_X\colon X\to Q_X$ and $ p_Y\colon Y\to Q_Y$ be the quotient mappings. When is there an equivariant biholomorphism of $ X$ and $ Y$? A necessary condition is that the categorical quotients $ Q_X$ and $ Q_Y$ are biholomorphic and that the biholomorphism $ \phi $ sends the Luna strata of $ Q_X$ isomorphically onto the corresponding Luna strata of $ Q_Y$. Fix $ \phi $. We demonstrate two homotopy principles in this situation. The first result says that if there is a $ G$-diffeomorphism $ \Phi \colon X\to Y$, inducing $ \phi $, which is $ G$-biholomorphic on the reduced fibres of the quotient mappings, then $ \Phi $ is homotopic, through $ G$-diffeomorphisms satisfying the same conditions, to a $ G$-equivariant biholomorphism from $ X$ to $ Y$. The second result roughly says that if we have a $ G$-homeomorphism $ \Phi \colon X\to Y$ which induces a continuous family of $ G$-equivariant biholomorphisms of the fibres $ p_X{^{-1}}(q)$ and $ p_Y{^{-1}}(\phi (q))$ for $ q\in Q_X$ and if $ X$ satisfies an auxiliary property (which holds for most $ X$), then $ \Phi $ is homotopic, through $ G$-homeomorphisms satisfying the same conditions, to a $ G$-equivariant biholomorphism from $ X$ to $ Y$. Our results improve upon those of our earlier paper [J. Reine Angew. Math. 706 (2015), 193-214] and use new ideas and techniques.


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Additional Information

Frank Kutzschebauch
Affiliation: Institute of Mathematics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Email: frank.kutzschebauch@math.unibe.ch

Finnur Lárusson
Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
Email: finnur.larusson@adelaide.edu.au

Gerald W. Schwarz
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110
Email: schwarz@brandeis.edu

DOI: https://doi.org/10.1090/tran/6797
Keywords: Oka principle, geometric invariant theory, Stein manifold, complex Lie group, reductive group, categorical quotient, Luna stratification.
Received by editor(s): April 23, 2015
Received by editor(s) in revised form: January 13, 2016
Published electronically: May 5, 2017
Additional Notes: The first author was partially supported by Schweizerischer Nationalfond grant 200021-140235/1
The first and third authors thank the University of Adelaide for hospitality and the Australian Research Council for financial support
The second author was partially supported by Australian Research Council grants DP120104110 and DP150103442
The second and third authors thank the University of Bern for hospitality and financial support
Article copyright: © Copyright 2017 American Mathematical Society